Cryogenic refrigeration technology is important for space infrared astronomical telescopes. This article summarizes the development history of cryogenic refrigeration technology for space infrared astronomical telescopes, and the developmental status of the cryogenic system for the Origins Space Telescope (OST) is reviewed. The OST is a planned mid- and far-infrared space telescope to explore the origin of the universe. To achieve normal in-orbit operation, a combination of mechanical cryocoolers and adiabatic demagnetization must be used to maintain a low-temperature working environment. This review and analysis of the development status of adiabatic demagnetization refrigerators (CARDs) and various mechanical cryocoolers (e.g., turbine Brayton cryocooler, multi-stage pulse tube cryocooler, and J-T cryocooler precooled by multi-stage cryocooler) at home and abroad, introducing their different characteristics, will provide some references for the development of our country’s infrared space telescopes in the future.